A priori error estimates of a fictitious domain formulation of the Newtonian cooling problem

Abstract We present a theoretical analysis of a fictitious domain formulation of the Newtonian cooling problem, motivated by applications in topology optimization. The method reformulates the classical heat conduction model with Robin-type boundary conditions on a fixed computational domain using a so-called weak material approximation. In this setting, the conductivity equals one in the solid subdomain $$\Omega _s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> and a small positive parameter $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> in the surrounding fictitious region. We derive a priori error estimates that quantify the consistency error between the extended and original formulations and prove that the solution restricted to $$\Omega _s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> converges to the true solution with an $$O(\epsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> error in the $$H^1(\Omega _s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> norm. We further provide $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -dependent finite element (FE) error estimates and show that for a mesh with characteristic size h , the condition number of the FE systems scales as $$O(\epsilon ^{-1}h^{-2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . To address the $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -induced ill-conditioning, we introduce a simple yet effective diagonal preconditioning strategy that removes the $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -dependency from the condition number. Numerical experiments confirm the theoretical convergence rate and demonstrate the effectiveness of the method. Thus, this work provides a theoretical foundation for utilizing weak material approximations for boundary-effect-dominated problems, thereby extending existing analyses to cases with Robin-type boundary conditions.